HAL Id: hal-00655963
https://hal.archives-ouvertes.fr/hal-00655963
Submitted on 3 Jan 2012HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
with logarithmic barriers
Pietro Caputo, Hubert Lacoin, Fabio Martinelli, François Simenhaus, Fabio
Toninelli
To cite this version:
Pietro Caputo, Hubert Lacoin, Fabio Martinelli, François Simenhaus, Fabio Toninelli. Polymer dy-namics in the depinned phase: metastability with logarithmic barriers. Probability Theory and Re-lated Fields, Springer Verlag, 2011, pp.Online first. �10.1007/s00440-011-0355-6�. �hal-00655963�
WITH LOGARITHMIC BARRIERS
PIETRO CAPUTO, HUBERT LACOIN, FABIO MARTINELLI, FRANC¸ OIS SIMENHAUS, AND FABIO LUCIO TONINELLI
Abstract. We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the de-pinned regime. At equilibrium the system exhibits a double well structure: the polymer lies (essentially) either above or below the repulsive line. As a consequence, one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However, the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. Our main results are: (i) a proof that the mixing time of the system lies between L52 and L52+2; (ii)
the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.
2000 Mathematics Subject Classification: 60K35, 82C20
Keywords: reversible Markov chains, polymer pinning model, metastability, spectral gap, mixing time, coupling, quasi-stationary distribution.
1. Introduction, model and results
Random polymers are commonly used in statistical mechanics to model a variety of inter-esting physical phenomena. A rich class of models with a non-trivial behavior is obtained by considering a simple random walk path interacting with a defect line in the thermodynamic limit when the length of the path tends to infinity. The equilibrium of these so-called polymer pinning models has been studied in depth in the mathematical literature, and the associated localization/delocalization phase transition is, nowadays, a well understood phenomenon, even in the presence of non-homogeneous interactions; see [9] for a recent survey.
Markovian stochastic dynamics of random pinned polymers, on the other hand, have received much less attention from a mathematical point of view. Besides its importance in bio-physical applications (see e.g. [5, 6] and references therein), the stochastic evolution of polymer models poses new challenging probabilistic problems from many points of view and the connection between the equilibrium and dynamical properties of the model is still largely unexplored. In particular, we feel that the problem of how the polymer relaxes to the stationary distribution (time scales, overcoming of energy barriers, metastability, patterns leading to equilibrium) still lacks a satisfactory solution even in the simplest homogeneous models; see [7] for some initial results in this direction.
In this paper we consider the dynamics of a homogeneous polymer model interacting with a repulsive defect line with two main motivations in mind:
(i) the repulsive regime is characterized by a relaxation to equilibrium occurring on a time scale certainly much larger [7] than the usual diffusive one which is typical e.g. of the neutral case1 [21]. The new scale is clearly the result of a subtle competition in the polymer
evolution between diffusive behavior and jumps against energy barriers.
(ii) The whole relaxation mechanism should show certain typical features of metastable evo-lution but in a very atypical context2 in which the relevant relaxation time scales are only
polynomial in the size L of the system (i.e. the energy barriers are only logarithmic in
L), with little separation between the mixing time inside one phase and the global mixing
time. A signature of this fact can be found in the anomalous growth with L of the global mixing time, a growth which is much more rapid than the naive guess based on the usual rule Tmix≈ exp(∆E), with ∆E the so-called activation energy. In order to appreciate the
novelty of such a situation it is useful to compare it to another well known case, namely the Glauber dynamics for the low temperature Ising model in a square box with free bound-ary [14], for which a very precise analysis of the metastable behavior was possible exactly because of a sharp separation, at an exponential level, between the two time scales. 1.1. Dynamics of the polymer pinning model. Let Ω = Ω2L denote the set of all lattice
paths (polymers) starting at 0 and ending at 0 after 2L steps, L∈ N:
Ω ={η ∈ Z2L+1: η−L= ηL= 0 , ηx+1 = ηx± 1 , x = −L, . . . , L − 1} .
The stochastic dynamics is defined by the natural spin-flip continuous time Markov chain with state space Ω. Namely, sites x = −L + 1, . . . , L − 1 are equipped with independent rate 1 Poisson clocks. When site x rings, the height ηx of the polymer at x is updated according to
the rules: if ηx−1 = ηx+1 ± 2 then do nothing; if ηx−1 = ηx+1 = h, and |h| ̸= 1, then set
ηx = h± 1 with equal probabilities; if ηx−1 = ηx+1 = +1, then set ηx = 0 with probability λ
λ+1 and ηx = 2 with probability
1
λ+1; similarly, if ηx−1 = ηx+1 = −1, then set ηx = 0 with
probability λ+1λ and ηx = −2 with probability λ+11 . Here λ > 0 is a parameter describing the
strength of the attraction (λ > 1) or repulsion (λ < 1) between the polymer and the line η ≡ 0. The infinitesimal generator of the Markov chain is given by
Lf(η) = L−1∑ x=−L+1 rx,+(η) [ f (ηx,+)− f(η)]+ L−1∑ x=−L+1 rx,−(η) [ f (ηx,−)− f(η)] , (1.1) where: f is a function Ω 7→ R; ηx,± denotes the configuration which coincides with η at every site y ̸= x and equals ηx± 2 at site x; the rates rx,± are zero unless ηx,± ∈ Ω, while if ηx,±∈ Ω
and ηx−1 = ηx+1 = h they satisfy rx,±= 12 for h̸= ±1, and rx,∓= λ+1λ = 1− rx,±, for h =±1.
The process defined above is the heat bath dynamics for the homogeneous polymer pinning
model, with equilibrium measure π = π2Lλ on Ω defined by
π2Lλ (η) = λ
N (η)
Z2Lλ , (1.2)
where N (η) = #{x ∈ {−L + 1, . . . , L − 1} : ηx = 0} denotes the number of zeros in the path
η ∈ Ω and Z2Lλ = ∑η′∈ΩλN (η′). For every λ > 0 and L ∈ N, π = π2Lλ is the unique reversible invariant measure for the Markov chain.
1In the neutral case (absence of an interaction between the polymer and the line) our process is nothing but
the usual (finite) symmetric simple exclusion model.
2Over the years there have been many different formulations of “metastability”; see [18, 20, 4]. We also refer
to the recent contributions [2, 3] where, as in our case, energy barriers are only logarithmic in the characteristic size of the system. We feel however that our situation does not fit completely in any of the mentioned contexts.
1.2. Relaxation to equilibrium. The equilibrium properties of the polymer pinning model have been studied in detail, cf. e.g. [10] or [9, Section 2] for an extensive review. In particular, it is well known that, under the measure π2Lλ , for λ > 1 the path is strongly localized with a non-vanishing density of zeros, while for λ < 1 the path is delocalized with √L height fluctuations
and with the number of zeros stochastically dominated by a geometric random variable with λ-dependent parameter. The dynamical counterpart of this localization/delocalization transition has not been fully understood yet. Some progress in this direction has been reported in [7], where various bounds on the spectral gap and mixing time of the Markov chain were obtained, together with estimates on the decay of time correlations.
We recall that the spectral gap is the smallest nonzero eigenvalue of −L, and one is often interested in the relaxation time Trel= 1/gap which governs decay to equilibrium inL2(π), while
the mixing time Tmix(δ), for δ∈ (0, 1), is the smallest time t such that
max
η ∥Pt(η,·) − π∥ 6 δ , (1.3)
where Pt(η,·) denotes the distribution of the Markov chain at time t with initial state η ∈ Ω, and
∥µ − ν∥ is the usual total variation distance between two probability measures. When δ = 1 2e we
often write simply Tmix instead of Tmix(δ). With these conventions one has Trel 6 Tmix always,
and the inequality is strict in general.
A dynamical phase transition occurs when we move from the localized regime λ > 1 to the delocalized regime λ < 1. It was shown in [7], see Theorem 3.4 and Theorem 3.5 there, that for
λ> 1 one has Trel= O(L2) and Tmix = O(L2log L), while for λ < 1 one has Trel> L
5
2−ε, (1.4)
for all ε > 0, provided L is large enough.
We refer to [7] for results and conjectures concerning the localized regime λ > 1. Here, we consider the delocalized regime, i.e. in the repulsive case λ < 1. The first question we address concerns an upper bound on the relaxation time Trel and the mixing time Tmix. It is worth
noting that even a crude polynomial bound is non-trivial. We refer to [15, 7] for polynomial bounds for the model with a horizontal wall at level zero, i.e. when lattice paths are constrained to be non-negative. On the other hand, without the wall constraint, the equilibrium measure π is known to be concentrated, as L → ∞, on configurations in which the density of monomers in the upper (lower) half plane is approximately one (zero). However, a mathematical working definition of the plus or minus phase for the polymer is not so obvious and we have been forced to introduce a mesoscopic parameter ℓ (i.e. L≫ ℓ ≫ 1) and define Ω± by
Ω+ ={η ∈ Ω : ηx> 0 , −L + ℓ < x < L − ℓ} , Ω− =−Ω+, (1.5)
where for any set A of polymer configurations −A := {η ∈ Ω : −η ∈ A}.
The presence of the two phases associated to Ω±dramatically changes the relaxation scenario, with a bottleneck at the set Ω\ (Ω+∪ Ω−). As explained in [7, Section 6], one may suspect that
Trel∼ L52 is the correct asymptotic behavior in the delocalized regime. Let us briefly recall the
heuristic reasoning behind this prediction.
The time to reach equilibrium can be roughly thought of as the time needed to switch from, say, Ω− to Ω+. A point x such that ηx = 0 and ηx−1 ̸= ηx+1 is called a crossing of the polymer.
Note that any zero (and therefore any crossing) x must belong to the set EL of points in the
segment{−L, . . . , L} which have the same parity as L. Since there are typically very few zeros at equilibrium, one may consider the extreme case where at most one crossing ξ is allowed at all
L L
L −L −L
−L ξ ξ ξ
Figure 1. From left to right a snapshot sequence of the motion of a single crossing ξ which allows the system to switch from a mostly negative to a mostly positive configuration.
times. In this case, the time evolution of ξ should be essentially described by a suitable birth and death process or random walk on EL; see Figure 1.
From equilibrium considerations, one knows that this random walk should have reversible invariant measure ρ roughly proportional to
ρ(x)∝ L3/2(L + x)−3/2(L− x)−3/2, x∈ EL, (1.6)
and that its relaxation time can be bounded from above and below by constant multiples of
L5/2; see Lemma 2.2 and Lemma 4.6 below for more details. Notice that, although the measure
ρ gives uniformly (in L) positive mass to the two attractors x± =±L, the drift which pushes the random walk away from the saddle x = 0 is proportional to the inverse of the distance from the attractors.
This heuristics is turned into the rigorous bound (1.4) by using a suitable test function in the variational principle that characterizes the spectral gap; see [7, Section 6]. However, it seems very hard to give a rigorous upper bound on Trel of the same order of magnitude. We obtain a
bound that can be off by at most two powers of L.
Theorem 1.1. For any λ < 1,
lim sup L→∞ log Tmix log L 6 5 2 + 2 .
The main tool for the proof of Theorem 1.1 is the analysis of an effective dynamics for the crossings of the polymer. To describe this, we introduce the variable σ ∈ {−1, +1}OL, where
OL={−L, . . . , L} \ ELdenotes the sites with the same parity of L + 1. If η is a configuration of
the polymer, then ηx ̸= 0 at any x ∈ OL, and we define σ(η) by σx = sign(ηx). The projection
of π on S = {−1, +1}OL is then
ν(σ) = ∑
η : η∼σ
π(η) , (1.7)
where the sum is over all configurations η compatible with the signs σ. The field ν has non-trivial long range correlations. Consider the heat bath dynamics for the variables σ: sites x∈ OL are
equipped with independent rate 1 Poisson clocks; when site x rings we replace σx by σx′ where
the new sign σx′ is distributed according to the conditional probability ν(· | σy, y ̸= x), i.e.
the probability (1.7) conditioned on the value of σy, y̸= x. Denote by TrelS the corresponding
relaxation time. For this process, the exponent 5/2 can be shown to be optimal.
Theorem 1.2. For any λ < 1,
lim L→∞ log TrelS log L = 5 2.
The proof of Theorem 1.1 and Theorem 1.2 combines several different tools which play a prominent role in the analysis of convergence to equilibrium of Markov chains: decomposition methods, spectral gap analysis, comparison inequalities, and coupling estimates. An outline of the main steps of the proof is given at the beginning of Section 4.
1.3. Metastability. Recall the definition (1.5) of the two sets Ω±, and define the associated phases as the restricted equilibrium measures π± := π(· | Ω±), so that (cf. Section 2)
π −1
2(π
++ π−) = o(1) . (1.8)
The notation o(1) refers to asymptotics as L → ∞. In the thermodynamic limit, we expect relaxation to equilibrium within each phase to occur on time scales Trel± such that Trel ≫ Trel±,
while on a time scale proportional to Trel one should see the system jump from one phase to the
other according to i.i.d. exponentially distributed times. A strong indication of this metastable behavior comes from the following theorem. Below, we use η(t) to denote the state at time t of the Markov chain with generator (1.1).
Theorem 1.3. There exists a set S+ ⊂ Ω+ such that π(S+) = 1/2 + o(1), and that uniformly
in η ∈ S+ and uniformly in t > 0:
Pη(τ−> t)= e−t/(2Trel) + o(1) ,
where
τ−= inf{t > 0 : η(t) ∈ S−} , S−=−S+.
Here Pη stands for the law of the process with initial state η ∈ Ω. By symmetry, Theorem 1.3 also implies that uniformly in η∈ S−=−S+ and uniformly in t > 0,
Pη(τ+> t)= e−t/(2Trel) + o(1) ,
where τ+= inf{t > 0 : η(t) ∈ S+}.
Remark 1.4. From the proof of Theorem 1.3 it will be clear that the set S+ is increasing w.r.t. the natural partial order among polymer configurations defined in Section 2.5, so that in particular the maximal configuration (in the sequel denoted by ∧) is in S+.
If we define the renormalized process
ωs= 1 ( η(sTrel)∈ Ω+ ) − 1(η(sTrel)∈ Ω− ) ,
we expect that, starting from any configuration in Ω±, {ωs, s > 0} converges to the simple
two-state Markov chain with switching rate (from ±1 to ∓1) equal to 1/2, whose spectral gap equals 1. Such a strong uniform result seems very hard to obtain for our model. The difficulty is that, in contrast with familiar metastability results [18], here there is no clear-cut separation of time scales: while (1.4) and Proposition 2.6 below imply Trel ≫ Trel+, the ratio Trel/Trel+ is only
polynomially large in L. However, we do have a detailed description of the renormalized process when the initial condition is the maximal configuration. Namely, define the maximal element of Ω as ηmax=∧, i.e. ∧x = x + L for x6 0 and ∧x= L− x for x > 0, and let Tmix∧ (ε) denote the
first time t such that ∥Pt(∧, ·) − π∥ 6 ε.
Theorem 1.5. For any δ > 0, uniformly in t> L2+δ
Pt(∧, ·) − [ 1 + e−t/Trel 2 π ++1− e−t/Trel 2 π −] = o(1), (1.9) and uniformly in t> 0 νπ+ t − [ 1 + e−t/Trel 2 π ++1− e−t/Trel 2 π −] = o(1) (1.10)
where νtπ+ denotes the law at time t of the process started from the initial distribution π+. Moreover, for any ε∈ (0, 1/2) one has
Tmix∧ (ε) = Trellog
( 1 2ε
)
(1 + o(1)). (1.11)
Remark 1.6. Theorem 1.5 shows in particular that, when the dynamics is started from either
π+ or ∧, there is no cut-off phenomenon [13], i.e. the variation distance from equilibrium does not fall abruptly to zero, but rather does so smoothly (on the timescale Trel). That is another
signature of the metastable behavior of our system and it is in contrast with what one expects for the neutral or attractive case λ> 1.
One of the key features of metastability is that, once the system decides to jump from e.g.
S+ to S−, then it does so very quickly on the time scale of the mixing time. We verify that this is indeed the case for most starting configurations inside S+∪ S−.
Let T denote the random time spent outside S+∪ S− up to the hitting time of S−:
T :=
∫ τ−
0
1{η(s)∈(S+∪S−)c}ds. (1.12) From Theorems 1.3 and 1.5 one easily deduces that, for most initial conditions in S+, τ−≫ T :
Corollary 1.7. There exists a subset ˜S+ of the set S+ of Theorem 1.3 satisfying π( ˜S+) = 1/2 + o(1) such that, uniformly on η∈ ˜S+, T = o(τ−) in probability, i.e. there exists a sequence
δL tending to zero as L→ ∞ such that for every η ∈ ˜S+,
Pη[T > δ Lτ−
]
6 δL. (1.13)
Along the same lines of the proof of the Corollary, one can establish the weak convergence of the renormalized process ωsto the two-state Markov chain, provided that the initial configuration
is inside a suitable subset of ˜S+∪ ˜S− with almost full measure. We decided to omit details for shortness.
1.4. Organization of the paper. The rest of the paper consists of three sections. Section 2 starts with standard material and then proceeds with the introduction of some essential tools to be used in the proof of the main results, including general results for monotone systems that can be of independent interest. This section contains also some new results concerning the relaxation within one phase and the properties of the principal eigenfunction of the generator. The metastability results are discussed in Section 3. Here, we start with the proof of Theorem 1.5. In later subsections we develop the construction needed for the proof of Theorem 1.3. Finally, Section 4 proves Theorem 1.1 and Theorem 1.2. This section is broken into several subsections corresponding to the various steps of the proof. A high level description of the arguments involved is given at the beginning of the section.
Notational conventions. Whenever we write o(Lp) or O(Lp) for some p∈ R it is understood that this refers to the thermodynamic limit L → ∞. Also, we use the notation f(L) = Ω(Lp)
when there exists a constant c > 0 such that f (L)> c Lp for all sufficiently large L. For positive functions f, g, we use the notation f (L) ≫ g(L) whenever lim infL→∞f (L)/g(L) = +∞, and
f (L)∼ g(L) when limL→∞f (L)/g(L) = 1. Also, we write f ≍ g if there exists some constant
2. Some tools
We begin with some generalities about reversible Markov chains. Then, we recall the definition of the polymer dynamics and derive some consequences of monotonicity. Next, we give some estimates on convergence to equilibrium in the “plus” phase. Finally, we characterize in detail an eigenfunction of L with eigenvalue −gap.
2.1. Preliminaries. We will consider reversible continuous time Markov chains with finite state space X, defined by the infinitesimal generator L acting on functions f : X 7→ R,
[Lf](x) = ∑
y∈X
c(x, y)[f (y)− f(x)] , (2.1) where c(·, ·) is a bounded non-negative function on X × X satisfying π(x)c(x, y) = π(y)c(y, x), for a probability measure π on X. In the applications below, the rates c(x, y) will always be such that the Markov chain is irreducible and the reversible invariant measure π is positive on
X. We refer e.g. to [1, 13] for more details on reversible Markov chains.
Let νx
t =P(vtx ∈ ·) denote the law of the state vtx of the Markov chain at time t with initial
condition x∈ X. We shall investigate the rate of convergence of νtxto π. If the initial condition x is distributed according to a probability µ on X, we write νtµ=∑x∈Xµ(x)νtxfor the distribution at time t. As usual, one can associate a semi-group {Pt, t > 0} to the generator L in such a
way that [Ptf ](x) = [etLf ](x) =
∑
y∈Xνtx(y)f (y). We also use the notation Pt(x, y) = νtx(y),
and νtµ= µPt.
The mixing time of the Markov chain is defined by
Tmix(ε) = inf { t > 0 , max x∈X ∥ν x t − π∥ 6 ε } , (2.2) where ∥µ − ν∥ = 1 2 ∑ x |µ(x) − ν(x)|
is the total variation distance. We shall use the convention that Tmix stands for Tmix(2e1). It is
well known that with this notation one has
∥νx
t − π∥ 6 e−⌊t/Tmix⌋, (2.3)
for all t > 0, where ⌊a⌋ denotes the integer part of a > 0. The spectral gap and the relaxation time of the process are defined by
gap = min f :X7→R E(f, f) Varπ(f ) , Trel= 1 gap, (2.4) where for f : X 7→ R, E(f, f) = ∑ x∈X π(x)f (x)[−Lf](x) = 1 2 ∑ x,y∈X π(x)c(x, y)[f (y)− f(x)]2 (2.5) is the quadratic form of the generator, a.k.a. the Dirichlet form, while Varπ(f ) stands for the
variance π(f2)−π(f)2. Thus, gap is the lowest non-zero eigenvalue of−L. The following bound relating total variation distance and relaxation time is an immediate consequence of reversibility and Schwarz’ inequality:
∥νµ t − π∥ 6 1 2e −t/Trel√Var π(f ) , (2.6)
where f (σ) = µ(σ)/π(σ) and µ is a probability on X. Another standard relation between total variation and relaxation time is the identity
gap =− lim
t→∞
1
t log maxx,y ∥ν x t − ν
y
t∥. (2.7)
Combining (2.3), (2.7) and (2.6), one can obtain the following well known relations:
Trel6 Tmix6
(
1− log π∗)Trel, where π∗= min
x∈Xπ(x) . (2.8)
2.2. A general decomposition bound on the spectral gap. We shall need a continuous time version of a general decomposition bound obtained by Jerrum et al. [11]. Consider the continuous time reversible Markov chain defined by (2.1). Suppose the space X is partitioned in the disjoint union of subspaces X1, . . . , Xm, for some m∈ N and define the generators
[Lif ](x) =
∑
y∈X
ci(x, y)[f (y)− f(x)] , ci(x, y) = c(x, y)1(y∈ Xi) , x∈ Xi.
Then Li is the generator of the Markov chain restricted to Xi, its reversible invariant measure
being given by πi = π(· | Xi). Let λmin denote the minimum of the spectral gaps of the Markov
chains generated byLi, i = 1, . . . , m. Next, letL denote the infinitesimal generator defined by
[Lφ](i) = m ∑ j=1 ¯ c(i, j)[φ(j)− φ(i)] , for φ∈ Rm, where ¯ c(i, j) = ∑ x∈Xi, y∈Xj π(x| Xi) c(x, y) .
This defines a continuous time Markov chain on {1, . . . , m} with reversible invariant measure ¯
π(i) = π(Xi). Let ¯λ denote the gap of this chain. A straightforward adaptation of [11, Theorem
1] yields the following estimate.
Proposition 2.1. Define γ = maximaxx∈Xi ∑
y∈X\Xic(x, y) . Then, with the notation of (2.4), gap> min { ¯λ 3, ¯ λ λmin ¯ λ + 3γ } . (2.9)
2.3. Killed process and quasi-stationary distribution. Here we recall some standard facts about killed processes, their generators and quasi-stationary distributions for reversible Markov chains; we refer to [1] for an introduction. Given a reversible Markov chain with generator L as above and a subset Γ⊂ X, we consider the process killed upon entering Γ, with sub-probability law defined by
νtx,Γ(B) =Px(vt∈ B ; τΓ > t) , x∈ Γc, (2.10)
where B⊂ X, vt denotes the state of the Markov chain with generator L at time t, Px denotes
the law of the process started at x, and τΓ denotes the hitting time of the set Γ. The associated
semi-group PΓ
t is given by
[PtΓf ](x) = [etLΓf ](x) = ∑
y∈Γc
νtx,Γ(y)f (y) , x∈ Γc, (2.11) where the killed generator LΓ satisfies, for every x∈ Γc:
[LΓf ](x) = [L(f1Γc)](x) = [Lf](x) − ∑
y∈Γ
We assume that PtΓis irreducible. ThenLΓ is a negative definite, self-adjoint operator inL2(π),
and its top eigenvalue −γΓ is characterized by
γΓ= min f :X7→R, f 1Γ=0 ⟨ −LΓf, f⟩ π π(f2) = minf :X7→R, f 1Γ=0 E(f, f) π(f2) , (2.13)
where we use ⟨·, ·⟩π for the scalar product inL2(π), and E(f, f) is defined by (2.5).
Let gΓ denote the (unique, positive on Γc) eigenfunction ofLΓ associated to −γΓ. Extending gΓ to all x∈ X by setting gΓ(x) = 0 for x∈ Γ, one defines the quasi-stationary distribution νΓ, i.e. the probability on X given by
νΓ(y) = π(y)gΓ(y)
π(gΓ) , y∈ X . (2.14)
An equivalent characterization of νΓ is as the limit νΓ(B) = lim
t→∞P x(v
t∈ B | τΓ> t) , (2.15)
where B ⊂ X, and the chosen initial point x ∈ Γc is arbitrary. The fundamental property of the quasi-stationary distribution is that, starting from νΓ, the hitting time τΓ is exponentially
distributed with parameter γΓ:
PνΓ(τ
Γ> t) = e−γΓt, (2.16)
where PνΓ stands for the law of the process when the initial state is distributed according to
νΓ. Another way of expressing quasi-stationarity is νΓPtΓ = e−γΓtνΓ, for all t > 0. A general
property of γΓ (cf. Lemma 3.1 below) is that γΓ> π(Γ) gap.
2.4. Polymer model. Let Ω = Ω2L stand for the space of all lattice paths defined in the
introduction. A partial order in Ω is given by
η6 η′ ⇐⇒ ηx 6 ηx′ , x =−L, . . . , L . (2.17)
Given ζ, ξ ∈ Ω such that ζ 6 ξ we define the restricted space Ωζ,ξ of all paths η ∈ Ω such that
ζ 6 η 6 ξ. The dynamics is defined by the continuous time Markov chain with state space Ωζ,ξ, with infinitesimal generator Lζ,ξ given by (1.1) where the rates rx,±(η) are replaced by
rx,ζ,ξ±(η) = rx,±(η)1(ηx,±∈ Ωζ,ξ) (2.18)
This process is the heat bath dynamics associated to the probability measure πλ,ζ,ξ2L on Ωζ,ξ defined as in (1.2) with the normalization now given by
Z2Lλ,ζ,ξ = ∑
η′∈Ωζ,ξ
λN (η′). (2.19)
Equivalently, πλ,ζ,ξ2L = πλ2L(· | Ωζ,ξ). This is referred to as the polymer model with top/bottom constraints (ζ is the bottom, ξ is the top). For simplicity, when no confusion arises, we often omit the superscripts λ, ζ, ξ and the subscript L from our notation in what follows. We write
vtη for the state of the Markov chain at time t when the initial configuration is some η, and let
νtη denote its distribution. When the initial condition η is distributed according to a probability measure µ on Ω we write νtµ as in Section 2.1.
Note that the generatorLζ,ξcan be written in the form (2.1) by setting c(η, η′) = rζ,ξx,±(η)1(η′ =
ηx,±), and π = πλ,ζ,ξ2L is reversible. While this holds for every value λ > 0 of the parameter describing the strength of the interaction, we will only consider the case λ < 1 below, which corresponds to a strictly delocalized regime for the polymer.
The minimal path ∨ and maximal path ∧ for the order (2.17) are defined by ∨x =−x − L
for x 6 0, ∨x = −L + x for x > 0, and ∧ = −∨. Clearly, if ζ = ∨ and ξ = ∧, then Ωζ,ξ = Ω.
This case is referred to as the polymer model with no top/bottom constraint.
The following well known estimates will be often used in our proofs. We refer e.g. to [9, Section 2] for the proof of Lemma 2.2 below, as well as for other known properties of the delocalized equilibrium measure. Let Z2L = Z2Lλ denote the partition function (2.19) with no top/bottom
boundaries and write Z2L+ = Z2L+,λ for the partition function (2.19) with ξ = ∧ and ζ given by the minimal non-negative element of Ω, i.e. ζx = 0 if x∈ EL (x has the same parity as L) and
ζx = 1 if x∈ OL (x has opposite parity w.r.t. L); Z2L+ is the partition function of the polymer
with a horizontal wall at height zero. Recall that N = N (η) stands for the number of zeros in the path η lying strictly between −L and L. Considering reflections of the path between consecutive zeros one obtains
2Z2L+,λ= Z2Lλ/2. (2.20)
Lemma 2.2. Consider the polymer with no top/bottom constraint with λ∈ (0, 1). There exist
constants ci = ci(λ) > 0, i = 1, 2 such that
2−2LZ2Lλ ∼ c1L−3/2, (2.21)
and
π(N (η) > k)6 c2e−k/c2, ∀ k ∈ N . (2.22)
An immediate implication of (2.20) and (2.21) is that
2−2LZ2L+,λ∼ c+L−3/2, (2.23)
for some constant c+ > 0 as soon as λ < 2. Moreover, (2.21) and (2.23) imply the bounds π (ηy > 0 ∀y ∈ {−L, . . . , x} , and ηx = 0) ≍ π (ηx = 0)
≍ L3/2(L + x)−3/2(L− x)−3/2, (2.24)
for every x∈ EL.
2.5. Monotonicity. An important property satisfied by the Markov chains introduced above is the monotonicity with respect to the partial order (2.17). A convenient way of stating the monotonicity property is that there exists a coupling P of the trajectories of the Markov chains corresponding to distinct initial conditions such that if η6 η′ thenP almost surely vηt 6 vηt′ for all t > 0. More generally, one can define a coupling P of trajectories corresponding to distinct top/bottom constraints and distinct initial conditions such that if ζ 6 ζ′, ξ 6 ξ′, and η 6 η′, thenP almost surely vη;ζ,ξt 6 vtη′;ζ′,ξ′ for all t> 0. Recall that a function f : Ω 7→ R is said to be increasing if f (η)6 f(η′) whenever η 6 η′. An event A is increasing if the indicator function 1A
is increasing. The monotonicity property of the dynamics implies the so-called FKG property of the equilibrium measures π = π2Lλ,ζ,ξ: for every pair of increasing functions f, g : Ω 7→ R, one has π(f g)> π(f)π(g). We refer to [7, Section 2] for a more detailed discussion of the monotone coupling and the consequences of monotonicity.
Lemma 2.3. Let µ be a probability on Ω and write f (η) = µ(η)/π(η), and ft(η) = νtµ(η)/π(η),
t > 0. If f is increasing then, for every t > 0, ft is increasing. As a consequence, there exists
an increasing event A such that ∥νµ
t − π∥ = ν µ
Proof. Write νtµ(η) = ∑η0∈Ωµ(η0)Pt(η0, η), where Pt(·, ·) stands for the kernel of the Markov
chain. Reversibility then gives
ft(η) = ∑ η0∈Ω f (η0)π(η0)Pt(η0, η)/π(η) = ∑ η0∈Ω f (η0)Pt(η, η0) . (2.26)
Next, let P denote the monotone coupling introduced above and let E denote expectation w.r.t. P. Then, (2.26) coincides with E[f(vη
t)], and if η6 η′, ft(η′)− ft(η) =E[f(vη ′ t )− f(v η t)] =E[f(v η′ t )− f(v η t) ; v η′ t > v η t] .
Thus, ft is increasing whenever f is. Finally, it is well known that the total variation distance
can be written in the form (2.25) where A = {η : νtµ(A) > π(A)}. Since A = {ft > 1}, A is
increasing whenever f is.
Lemma 2.4 compares arbitrary initial conditions to the extremal initial conditions. Lemma 2.5 states a useful sub-multiplicativity property satisfied by extremal evolutions. For lightness of notation, we state these results only in the case of no top/bottom boundaries, i.e. ζ =∨, ξ = ∧, but the same applies for general ζ, ξ with exactly the same proof.
Lemma 2.4. For any t > 0 and any η, η′ ∈ Ω:
∥νη t − ν η′ t ∥ 6 4L2∥νt∧− νt∨∥ . As a consequence, gap =− lim t→∞ 1 tlog∥ν ∧ t − νt∨∥ .
Proof. Let P denote the monotone coupling as above. Then, ∥νη t − ν η′ t ∥ 6 P(v η t ̸= v η′ t )6 P(v∧t ̸= vt∨) 6 L−1∑ x=−L+1 L−1∑ h=−L [P((vt∧)x > h)− P((vt∨)x> h)] 6 4L2∥ν∧ t − νt∨∥ .
The second point follows from the first one and the classical characterization (2.7) of the spectral
gap.
Lemma 2.5. For any s, t> 0,
∥νt+s∧ − νt+s∨ ∥ 6 ∥νt∧− νt∨∥ ∥νs∧− νs∨∥ .
Proof. With the same argument of Lemma 2.3, for some increasing event A ∥νt+s∧ − νt+s∨ ∥ = νt+s∧ (A)− νt+s∨ (A) .
Let ρ be a coupling beween νt∧ and νt∨ at fixed time t> 0. Then
νt+s∧ (A)− νt+s∨ (A) = ∫ (νsη(A)− νsσ(A))dρ(η, σ) = ∫ (νsη(A)− νsσ(A))1(σ̸= η)dρ(η, σ) 6 (νs∧(A)− νs∨(A))ρ(σ̸= η) 6 ∥νs∧− νs∨|| ρ(σ ̸= η) .
2.6. Relaxation in one phase. Here we obtain results concerning the polymer dynamics in the phase π+= π(·|Ω+) with Ω+ defined after (1.5); cf. Proposition 2.6 below. Then, we show that the polymer started at the maximal configuration ∧ relaxes first to the restricted equilibrium
π+ in a time O(L2+δ) for arbitrarily small δ > 0, while it takes much longer to reach the full equilibrium π; cf. Lemma 2.7 and Lemma 2.8 below.
Recall the definition (1.5) of the subspace Ω+ ⊂ Ω, where L ≫ ℓ, and ℓ diverges as L → ∞; see (2.29) below. The corresponding restricted equilibrium is given by π+= π(·|Ω+). Note that this is a particular instance of the polymer equilibrium πλ,ζ,ξ2L with top/bottom boundaries: the top is ξ = ∧ while the bottom ζ = ζ(Ω+) is the lowest element of Ω+. Similarly, one defines Ω− = −Ω+, i.e. use (1.5) with ηx > 0 replaced by ηx < 0, and the equilibrium π− is defined
accordingly.
Since λ < 1, the equilibrium bounds (2.24) imply that
π(Ω+) = π(Ω−) = 1 2+ O(ℓ
−1/2) ; (2.27)
see e.g. [7, Section 2]. In particular, if ℓ diverges as L→ ∞, then π−1
2(π
++ π−) = o(1) . (2.28)
What follows depends only marginally on the precise dependence of ℓ on L, provided that
L≫ ℓ ≫ 1. For the sake of simplicity we shall fix its value as
ℓ(L) =⌊(log L)14⌋ . (2.29)
This choice turns out to be convenient in the proof of Proposition 2.6 below, but we point out that any choice of the form ℓ(L) = O(Lε) for small ε > 0 would be sufficient to obtain the same conclusion with a little more work.
We start by establishing a mixing time upper bound for the dynamics constrained to stay in Ω+, i.e. the process evolving with bottom boundary given by ζ = ζ(Ω+). To avoid confusion we shall write µηt (instead of νtη) for the law at time t of this Markov chain with state space Ω+and
initial condition η. We write L+ for its generator and gap+ for the associated spectral gap.
Proposition 2.6. For every ε > 0, there exists L0 = L0(ε) such that for all L > L0, for all t> 0 and all initial conditions η ∈ Ω+:
∥µη
t − π+∥ 6 4L2exp
(
− t/L2+ε). (2.30)
In particular, gap+> L−2−ε.
Proof. The last statement follows from (2.30) and (2.7). To prove (2.30) we establish that for
every ε > 0, there is a constant L0 = L0(ε) > 0 such that, taking T = L2+ε, we have ∥µ∧T − µ
ζ
T∥ 6 1 − L−ε, (2.31)
for all L > L0(ε), where ζ stands for the minimal element ζ = ζ(Ω+) of Ω+. Once (2.31) is
available, we obtain (2.30) (with a new value of ε) from Lemma 2.4, since Lemma 2.5 (which is also valid for the restricted dynamic) and (2.31) imply
∥µ∧t − µ ζ t∥ 6 ( ∥µ∧T − µ ζ T∥ )⌊t/T ⌋ 6 exp(−⌊t/T ⌋L−ε))6 2 exp(− t/L2+2ε),
for any L large enough.
To prove (2.31), we divide the sites x∈ {−L, . . . , L} in three overlapping regions:
where ℓ = ℓ(L) is given by (2.29). Let T2 = L2+ε1, T1 = Lε1 with some ε1 > 0 such that T = L2+ε > T′ := 2T2+ T1. We shall prove (2.31) with T replaced by T′ (this implies the
claim since by Lemma 2.4 the left hand side of (2.31) is monotone as a function of T ). Call
µη0,c
T′ the law of the “censored” process obtained as follows. Start from η0 at time 0 and, for
time t ∈ [0, T2] reject all the updates involving x /∈ I2. For time t ∈ (T2, T2 + T1] reject all
updates involving x /∈ I1 ∪ I3, and for time t∈ (T2+ T1, 2T2+ T1 = T′] reject all the updates
involving x /∈ I2. From the Peres-Winkler censoring inequality [19, Theorem 16.5] one has that
µ∧,cT′ stochastically dominates µ∧T′. Similarly, µ ζ,c
T′ is stochastically dominated by µ ζ
T′. On the
other hand, as in Lemma 2.3, one has
∥µ∧T′ − µ ζ
T′∥ = µ∧T′(A)− µ ζ T′(A) ,
where A⊂ Ω+ is an increasing event. Therefore,
∥µ∧ T′− µ ζ T′∥ 6 µ∧,cT′ (A)− µ ζ,c T′(A)6 ∥µ∧,cT′ − µ ζ,c T′∥ ,
and the lemma follows once we show that
∥µ∧,cT′ − µζ,cT′∥ 6 1 − L−ε. (2.32) To prove (2.32) we shall couple the two configurations ηT∧,c′ , ηTζ,c′ with law µ∧,cT′ , µζ,cT′ respectively. From the analysis of the polymer with the wall [7, Section 4], it is not hard to infer that uniformly in the boundary values at −L + ℓ − 1 and L − ℓ + 1 the system evolving in the region I2 has a mixing time O(L2log L). Therefore, after a time T2 = L2+ε1, up to O(L−p) corrections for a
large constant p > 0, for any event E, µ∧,cT
2(E) coincides with the equilibrium probability of E
in I2 with boundary conditions ∧ at −L + ℓ − 1 and L − ℓ + 1. The same applies to µζ,cT2(E)
provided the equilibrium is taken with boundary conditions ζ at−L+ℓ−1 and L−ℓ+1. Choose the event E that the configuration is minimal (in Ω+) at both−L + ℓ2− 1 and L − ℓ2+ 1 (i.e.
η−L+ℓ2−1 = ηL−ℓ2+1 = 0 if L− ℓ2 is odd, and η−L+ℓ2−1 = ηL−ℓ2+1 = 1 if L− ℓ2 is even). From
known equilibrium estimates [9, Section 2], it is not difficult to show that at equilibrium, with either of the two boundary conditions considered above, the probability of E is bounded below by
c1ℓ−6 for some constant c1> 0 depending only on λ. Therefore, using an independent coupling
in the time-lag [0, T2], we have that the event E occurs for both ηT∧,c2 , ηTζ,c2 with probability at
least c ℓ−12.
Next, conditioned on the event E we see that from time T2 up to time T2 + T1 the two
processes evolve (in the regions I1 and I3 only) with the same boundary conditions (equal to the
minimal configuration at both −L + ℓ2 and L− ℓ2). Since T1 = Lε and the mixing time of the
system in the regions I1 and I3 (which evolve independently) is certainly at most eO(ℓ
2)
, with probability 1 + O(L−p) (conditionally on event E) we have that the two configurations coincide in both regions I1, I2 at time T2+ T1. Therefore if we let the system run only in I2 now for an
additional time T2 we have a probability close to 1 to have the two configurations coinciding
everywhere. It follows that, conditionally on the event E, there is a probability of, say, at least 1/2 of no discrepancy between ηT∧,c′ and ηTζ,c′. Therefore, lettingP denote the coupling described
above, ∥µ∧,cT′ − µζ,cT′∥ 6 P(η∧,cT′ ̸= ηTζ,c′) 6 P(η∧,cT′ ̸= η ζ,c T′ | E)P(E) + 1 − P(E) 6 1 −1 2P(E) .
We now go back to the model with no top/bottom boundaries, that is the law νtη corresponds to the evolution with ξ =∧, ζ = ∨. The next result is crucially based on estimates obtained in [7, Section 6] for the delocalized regime λ < 1.
Lemma 2.7. Uniformly in t6 L5/2(log L)−9,
νt∧(Ω+) = 1 + o(1) .
Proof. Define the event A ={ ∑Lx=−Lηx < L3/2(log L)−3 }. Proposition 6.2 in [7] proves that
νt∧(A) = o(1) uniformly in t6 L5/2(log L)−9. Since Ω−⊂ A we have
νt∧(Ω−) = o(1) , t6 L5/2(log L)−9. (2.33) Next, let us check that
νt∧(Ω+| (Ω−)c)> π(Ω+| (Ω−)c) , t> 0 . (2.34) To this end, observe that since (Ω−)c is increasing, using Lemma 2.3 the function
f (σ) = 1(Ω−)c(σ) νt∧(σ| (Ω−)c) π(σ| (Ω−)c) = 1(Ω−)c(σ) νt∧(σ) π(σ) π((Ω−)c) νt∧((Ω−)c),
is increasing. Since Ω+⊂ (Ω−)c is increasing, with the FKG property for π, this implies (2.34). From (2.33) and (2.34) we obtain
νt∧(Ω+)> (1 + o(1))νt∧(Ω+| (Ω−)c)> (1 + o(1))π(Ω+| (Ω−)c) = 1 + o(1) ,
where the last bound follows from (2.27).
Lemma 2.8. For any ε > 0, uniformly in t∈ [L2+ε, L5/2(log L)−9] : ∥νt∧− π+∥ = o(1) ,
where π+ is defined by π+= π(· | Ω+).
Proof. Using Lemma 2.7 it is enough to prove
∥νt∧(· | Ω+)− π+∥ = o(1) ,
uniformly in t∈ [L2+ε, L5/2(log L)−9]. Consider the function f : Ω7→ R given by
f (σ) = 1Ω+(σ) νt∧(σ| Ω+) π+(σ) = 1Ω+(σ) νt∧(σ) π(σ) π(Ω+) νt∧(Ω+).
Since Ω+ is increasing, Lemma 2.3 shows that f is increasing. Therefore, the event
A ={σ ∈ Ω+ : νt∧(σ| Ω+) > π+(σ)} is increasing. Using monotonicity we have
νt∧(A| Ω+) = νt∧(A)/νt∧(Ω+)6 µ∧t(A)/νt∧(Ω+) ,
where µ∧t denotes the evolution constrained to stay in Ω+; see Proposition 2.6. Therefore, ∥ν∧
t(· | Ω+)− π+∥ = νt∧(A| Ω+)− π+(A)6
µ∧t(A)
νt∧(Ω+) − π +(A) .
The conclusion now follows from Lemma 2.7 and Proposition 2.6. The full power of Lemma 2.8 will be seen in the next sections. One of its consequences is the fact that the mixing time Tmix can be bounded in terms of the relaxation time via
for some constant c > 0. Indeed, (2.35) follows quite easily from Lemma 2.4, Lemma 2.8 and (2.6); see Lemma 4.1 below for a more subtle application of the same reasoning. Note that, since
Trel≫ L5/2−ε, the bound (2.35) improves considerably the standard estimate (2.8) by replacing the factor− log π∗ = O(L) with a factor O(log L).
2.7. Characterization of the principal eigenfunction. What follows refers to the model with no top/bottom boundaries. Recall that ∧, (resp. ∨) denotes the maximal (resp. minimal) configuration in Ω. A function g : Ω7→ R is called antisymmetric if g(−η) = −g(η) for all η ∈ Ω. The following result gives a precise characterization of one eigenfunction corresponding to−gap.
Proposition 2.9. There exists an increasing antisymmetric eigenfunction g of L, such that
∥g∥L2(π)= 1. It satisfies
Lg = −gap g. (2.36)
Moreover, when L tends to infinity
g(∧) = ∥g∥L∞ = 1 + o(1)
and
∥g − (1Ω+ − 1Ω−)∥L
1(π)= o(1). (2.37)
Proof. By decomposing 1∧− 1∨ on a basis of eigenfunctions of L one sees that
g := lim
t→∞
Pt(1∧− 1∨)
∥Pt(1∧− 1∨)∥L2(π)
is an eigenfunction with unit L2 norm. It is increasing and antisymmetric as Pt(1∧− 1∨) is
antisymmetric and increasing for all t (Pt preserves monotonicity and symmetries). To prove
(2.36), it suffices to show that the projection of 1∧− 1∨ on the eigenspace of L associated to
−gap is non-zero. To do so, first observe that by reversibility,
1
π(∧)∥Pt(1∧− 1∨)∥L1(π)= 2∥ν
∧
t − νt∨∥.
Then, by the second point of Lemma 2.4 lim t→∞ 1 tlog∥Pt(1∧− 1∨)∥L2(π)= limt→∞ 1 t log∥Pt(1∧− 1∨)∥L1(π) = lim t→∞ 1 t log∥ν ∧ t − νt∨∥ = −gap ,
(where we used equivalence of the norms in finite dimensional spaces).
We now estimate the L∞ norm of g. Let ε > 0 be small and fixed, and let t0 be such that L2+ε < t0 < L5/2−ε. The function g is an eigenfunction for Pt0 = exp(t0L), with eigenvalue
e−t0gap. Therefore e−t0gapg(∧) = E[g(v∧ t0) ] 6 π+(g) + 2g(∧)∥ν∧ t0 − π +∥,
where the last inequality follows from the fact that for any two measures µ, ν and any function
f ,
|µ(f) − ν(f)| 6 2∥f∥L∞∥µ − ν∥ . Hence, by Lemma 2.8 and the fact that gap−1 ≫ t0 (cf. (1.4)):
g(∧) 6 π +(g) e−t0gap− 2∥ν∧ t0− π +∥ = π +(g)(1 + o(1)). (2.38)
Moreover, by symmetry and Jensen’s inequality 1 = π(g2)> 2 ∑ η∈Ω+ π(η)g(η)2= 2π(Ω+)π+(g2)> 2π(Ω+)π+(g)2, so that π+(g)6 (2π(Ω+))−1/2= 1 + o(1).
Therefore g(∧) 6 1 + o(1) (and it is trivial to notice that ∥g∥L∞ > ∥g∥L2(π)= 1).
We turn to the proof of (2.37). First notice that by (2.38) one has π+(g)> (1+o(1))g(∧) > 1+
o(1) so that
π+(g) = 1 + o(1). (2.39)
Next, we prove that the variation of g within Ω+ is small:
Varπ+(g) = o(1). (2.40)
Indeed, Varπ+(g) = π+(g2)− π+(g)2 6 g(∧)2− π+(g)2 and the claim follows from (2.38). Next,
∥g − 1Ω++ 1Ω−∥L
1(π)6 ∥1Ω+(g− 1)∥L1(π)+∥1Ω−(g + 1)∥L1(π)+∥g1Ω\(Ω+∪Ω−)∥L1(π).
The first two terms of the right-hand side are equal by symmetry. Adding and subtracting
π+(g), and using Schwarz’ inequality,
∥1Ω+(g− 1)∥L 1(π)6 π(Ω +)[|π+(g)− 1| +√Var π+(g) ] = o(1) ,
where the conclusion follows from (2.39) and (2.40). The third term ∥g1Ω\(Ω+∪Ω−)∥L1(π) is
smaller than ∥g∥L∞π(Ω\ (Ω+∪ Ω−)) = o(1).
3. Metastability
In this section we first prove Theorem 1.5, which is mainly a consequence of the technical lemmas of the previous section and then move to the proof of Theorem 1.3 and its corollary. 3.1. Proof of Theorem 1.5. We use the notation T = L2+δ. Equation (1.11) is an easy consequence of (1.9) and (2.28). Indeed, assuming (1.9), for t> T , one has
∥Pt(∧, ·) − π∥ = [1 + e−t/Trel 2 π + + 1− e−t/Trel 2 π −]− π + o(1) = 1 2e −t/Trel + o(1). (3.1)
To prove (1.9), one writes νt∧− [1 + e−t/Trel 2 π ++1− e−t/Trel 2 π −] 6 νT∧Pt−T − π+Pt−T + π+Pt−T − [1 + e−t/Trel 2 π ++1− e−t/Trel 2 π −] . (3.2)
This is just triangular inequality, combined with the observation that νT∧Pt−T = νt∧. The first
term on the right hand side is smaller than ∥νT∧− π+∥ (as P
t−T contracts the norm) which is
itself small, by Lemma 2.8 and the definition of T . It remains to estimate the second term i.e. to prove (1.10).
To do this, we use the fact that the density of π+ w.r.t. π is very close to g + 1, where g is the eigenfunction described in Proposition 2.9, so that the density of π+Pt must be close to
Pt(g + 1). Using reversibility, one can express the densities as follows: dπ +P t dπ = Pt dπ+ dπ . Then we
rewrite the second term in (3.2) as an L1 norm (omitting a harmless factor 1/2)
Pt−T dπ+ dπ − 1 2π(Ω+)(1Ω++ 1Ω−)− 1 2π(Ω+)e−t/T rel(1 Ω+− 1Ω−) L1(π) 6 Pt−T dπ+ dπ − 1 2π(Ω+)(1Ω+ + 1Ω−)− 1 2π(Ω+)e−t/T relg L1(π) + 1 2π(Ω+)∥e −t/Trel(1 Ω+ − 1Ω−− g) ∥L1(π).
The last term above is small by Proposition 2.9. From (2.27) we know that 2π(Ω+) = 1 + o(1).
One can then estimate the first term Pt−Tdπ + dπ − 1 2π(Ω+)(1Ω++ 1Ω−)− 1 2π(Ω+)e−t/T relg L1(π) 6 ∥ (1Ω++ 1Ω−)− 1∥L1(π)+ Pt (dπ+ dπ − 1 − g ) L1(π) + o(1) , (3.3) where we used the triangular inequality, the fact that Pt1 = 1, and
Pt−Tg = e−(t−T )/Trelg = (e−t/Trel + o(1))g ,
which follows from T = o(Trel). On the right hand-side of (3.3), the first term is small by (2.28)
and the second is bounded by ∥dπdπ+ − 1 − g∥L1(π), which is small by Proposition 2.9. 3.2. Proof of Theorem 1.3. Theorem 1.5 gives some intuition on why the result should be true, and it will be used to determine the time of the jump from one state to the other. However, one needs another key ingredient to get the result, namely the description of the quasi-stationary distribution. The reason for this is that starting from the quasi-stationary distribution, a killed process dies exactly at exponential rate; see Section 2.3. Therefore, most of our effort will fo-cus on stochastic comparison with the quasi-stationary distribution. Let us first give a brief roadmap to help the reader through the proof of Theorem 1.3.
Step 1. The sets S± of Theorem 1.3 for which we have the desired exponential hitting time description are constructed by successively refining a first attempt. One first defines S0,±as the sets of polymer configurations where the eigenfunction g in Proposition 2.9 is positive (negative) and one verifies that their equilibrium probability is 12 + o(1). Then one examines the Dirichlet problem associated to the process killed in S0,− (S0,+) and one proves that the corresponding eigenvalue γ0is of the same order as the spectral gap apart from a crucial unspecified
multiplica-tive factor in [1/2, 1]. Similarly one verifies that the corresponding quasi-stationary measure is very close to the equilibrium measure π conditioned to be in S0,+(S0,−). In this way we get the exponentiality of the hitting time of e.g. S0,− starting from π+ with a rate which is, modulo a multiplicative factor in [1/2, 1], the spectral gap (see Lemma 3.3).
Step 2. Next one appropriately defines new sets S1,±⊂ S0,±in order to guarantee that this time the corresponding Dirichlet eigenvalue γ1 is equal to (12+ o(1))gap, and that the hitting time of S1,∓ starting from equilibrium conditioned to S1,± is exponential (with the correct rate). Again one of the key points is to show that π+ is close to the quasi-stationary distribution associated the process killed on entering S1,−, and that the equilibrium probability of S1,± is still 12+ o(1).
Step 3. Finally, one defines the sets S2,± ⊂ S1,± in such a way that: a) the hitting time of
S2,∓ starting from any configuration in S2,± (and not just from the conditional equilibrium) is also exponential with the correct rate 12gap; b) the equilibrium probability of S2,±is still 12+o(1). It is now time to begin the implementation of the above strategy. Let
S0,+:={η ∈ Ω, g(η) > 0} , (3.4) where g is the eigenfunction defined by Proposition 2.9 and set S0,− =−S0,+. From Proposi-tion 2.9,
∥1S0,+− 1Ω+∥L
1(π)= o(1). (3.5)
In particular, π(S0,±) = 1/2 + o(1). Let S− ⊂ S0,− be a decreasing event. We consider the
quasi-stationary distribution ν+ := νS− of the process killed when it hits S−. Let Pt∗ = PS
−
t ,
resp. L∗ = LS−, denote the semi-group, resp. the generator, associated to this process (see Section 2.3), −γS− =−γ be the largest eigenvalue of L∗ and τ− = τS− be the hitting time of
S−. From (2.16):
Pν+
(τ−> t) = e−γt. (3.6) Our first step is to prove that if S− has non-negligible measure, then γ is of the same order of the gap. More precisely:
Lemma 3.1. For any S−⊂ S0,−, one has π(S−)6 γ Trel 6 1.
Proof. The bound π(S−) 6 γ Trel is rather standard, but we include its proof for the sake of
completeness. Let f0 = gS− denote the minimizer in the variational principle defining γ = γS−;
see (2.13). Then Varπ(f0) =⟨f0, f0⟩π− ⟨ f0, 1(S−)c ⟩2 π > ⟨f0, f0⟩ππ(S−) ,
where we used the Cauchy-Schwarz inequality for ⟨f0, 1(S−)c ⟩2 π. Therefore γ = E(f0, f0) π(f02) > π(S −)E(f0, f0) Varπ(f0) > π(S −) gap .
As for the bound γ Trel 6 1, γS− being a non-decreasing function of S− (for the inclusion),
it is sufficient to prove the result for the maximal case S− = S0,−. Let g be the eigenfunction defined in Proposition 2.9. From (2.12), for all η∈ (S0,−)c
−(L∗g|
(S0,−)c)(η) =−(Lg)(η) +
∑
η′∈S0,−
c(η, η′)g(η′)6 − (Lg)(η) = gap g(η) ,
where we use the fact that g(η′) < 0 for η′ ∈ S0,−. Plugging this into (2.13), and using
g| (S0,−)c > 0, one gets γ 6 ⟨ −L∗g |(S0,−)c, g|(S0,−)c ⟩ π π(g2 |(S0,−)c ) 6 gap . Next, we prove that the quasi-stationary distribution ν+ for the process killed on S− is very close to π+ if S− has probability close to 1/2.
Lemma 3.2. Uniformly for all decreasing events S−⊂ S0,−, ∥ν+− π+∥ 6 (2 − 4π(S−)) + o(1) .
Proof. We use triangular inequality to get
∥ν+− π+∥ 6 ∥ν+− π(· |(S−)c)∥ + ∥π+− π(· |(S−)c)∥. (3.7)
We start with the first term. First, from (2.15) one has the characterization
ν+= lim
t→∞
δ∧Pt∗ δ∧Pt∗(Ω).
Since the operator Pt∗preserves monotonicity (S−is decreasing), arguing as in [19, Lemma 16.6], the density d[δ∧Pt∗]
dπ is seen to be an increasing function for every fixed t> 0. Hence, passing to
the limit t→ ∞, dν+/dπ is an increasing function. Therefore, A := { η∈ (S−)c, such that ν +(η)π((S−)c) π(η) > 1 }
is an increasing event. From standard properties of the total variation distance
∥ν+− π(·|(S−)c∥ = ν+(A)− π(A |(S−)c).
We shall prove that ν+(A) is smaller than π+(A) + o(1) by the use of monotonicity and a chain of comparisons. Recall the notation T = L2+δ (δ ∈ (0, 1/4)). We first compare ν+ to ν+PT:
remark that
ν+PT = ν+PT∗+ ν+(PT − PT∗)
where the two terms of the decomposition are positive measures. From quasi-stationarity one has ν+PT∗ = e−γTν+ and therefore the total mass of the second term above is 1− e−γT. Hence
∥ν+P T − ν+∥ = 1 2 d[ν+(PT − PT∗)] dπ − dν+ dπ (1− e −γT) L1(π) 6 1 − e−γ T = o(1). (3.8) The last equality comes from Lemma 3.1 and the fact that Trel≫ T . Next, δ∧PT stochastically
dominates ν+PT so that [ν+PT](A)6 [δ∧PT](A). Hence, from Lemma 2.8 and (3.8):
ν+(A)6 ν+PT(A) + o(1)6 δ∧PT(A) + o(1)
6 π+(A) +∥δ
∧PT − π+∥ + o(1) = π+(A) + o(1).
Therefore, going back to (3.7)
∥ν+− π+∥ 6 ν+(A)− π(A |(S−)c) +∥π+− π(· |(S−)c)∥ (3.9)
6 π+(A)− π(A |(S−)c) +∥π+− π(· |(S−)c)∥ + o(1) 6 2∥π+− π(· |(S−)c)∥ + o(1) .
To estimate the right-hand side of (3.9), notice that
∥π+− π(· |(S−)c)∥ 6 ∥π+− π(· |(S0,−)c)∥ + ∥π(· |(S0,−)c)− π(· |(S−)c)∥ , (3.10)
and the first term is o(1) by Proposition 2.9. Moreover, since S−⊂ S0,−
∥π(· |(S0,−)c)− π(· |(S−)c)∥ = π((S−)c)− π((S0,−)c) π((S−)c) = 1/2− π(S −) + o(1) 1− π(S−) 6 1 − 2π(S −) + o(1) . (3.11)
Combining (3.9), (3.10) and (3.11), the desired result follows. Now one uses the fact that ν+ and π+ are close in total variation distance to estimate the jumping time to S− starting from either∧ or from π+. For the rest of this section, one defines, in analogy with τ−, the hitting times τi,− (resp. τi,+), (i = 0, 1, 2) of the sets Si,− (resp. Si,+) to be defined.
